Abstract: Consider a Kahler manifold (X, g). When g can be expanded in power series, in his seminal work on holomorphic isometries Eugenio Calabi introduced the notion of the diastasis and proved powerful extension theorems on holomorphic isometries from Kahler manifolds into space forms such as the projective space equipped with the Fubini-Study metric. On a bounded domain U in C^n we denote by ds^2_U the Bergman metric on U (which is Kahler). Among bounded domains there are the bounded symmetric domains Ω classified by Élie Cartan such that (Ω, ds2_Ω) are symmetric in the sense of Riemannian geometry, Ω = G/K in standard notation. Here Ω in C^n in their standard realizations are semi-algebraic, i.e., defined by algebraic inequalities in the 2n real Euclidean coordinates underlying C^n ≅R^2n. By an irreducible algebraic subvariety of Ω we mean an irreducible component of the intersection V ∩ Ω of an affine algebraic subvariety V ⊂ C^n with the bounded symmetric domain Ω in C^n. In this lecture I will explain: (1) how the study of holomorphic isometries between bounded domains was motivated by problems in arithmetic dynamics, (2) how their solutions were generalized to yield algebraicity results for holomorphic isometries with respect to the Bergman metric, (3) how the study of the asymptotic behavior of holomorphic isometries of the Poincare disk led to a uniformization theorem for projective varieties covered by algebraic subvarieties of Ω, and (4) how the latter serves as a starting point for research in functional transcendence theory concerning XΓ = Ω/Γ, where Γ ⊂ G is an arbitrary lattice. In the special case of arithmetic lattices, (4) has been settled yielding the Ax-Schanuel theorem on Shimura varieties (with extensive generalizations by now) by Mok-Pila-Tsimerman (2019), using techniques involving in particular model theory from mathematical logic, techniques which are no longer available for arbitrary lattices. In case Ω is of rank 1, i.e, Ω ≅ B^n, Mok (2019) introduced a method using rescaling and complex geometry to give a proof of the Ax-Lindemann Theorem for X = B^n/Γ for arbitrary lattices Γ ⊂ Aut(B^n). In the case where X = Ω/Γ and Γ ⊂ Aut(Ω) is an arbitrary cocompact lattice, I will explain how functional transcendence results concerning X can be proven using analytic techniques starting with the rescaling method on subvarieties of a bounded symmetric domain exiting ∂Ω.